D. Pelinovsky, G. Schneider, and R. MacKay

Justification of the lattice equation for a nonlinear elliptic problem with a periodic potential

We justify the use of the lattice equation (the discrete nonlinear Schrodinger equation) for the tight-binding approximation of stationary localized solutions in the context of a continuous nonlinear elliptic problem with a periodic potential. We rely on properties of the Floquet band-gap spectrum and the Fourier-Bloch decomposition for a linear Schrodinger operator with a periodic potential. Solutions of the nonlinear elliptic problem are represented by the Wannier functions and the problem is reduced, with the elliptic theory, to a set of nonlinear algebraic equations solvable with the Implicit Function Theorem. Our analysis is valid for a class of piecewise-constant periodic potentials with disjoint spectral bands, which reduce, in a singular limit, to a periodic sequence of infinite walls of a non-zero width. The discrete nonlinear Schrodinger equation is applied to classify localized solutions of the Gross-Pitaevskii equation with a periodic potential.

justification of amplitude equations, gap solitons in periodic potentials, Gross-Pitaevskii equation, Fourier-Bloch decomposition, Wannier functions, discrete nonlinear Schrodinger equation